For a hard$-$sphere model in $2$D $($consider the spheres as disks$)$ shown in the next page,

$($a$)$ Estimate the sphere diameter $R_o.$ Note the scale bar given.

$($b$)$ Estimate the volume fraction $V_F($i$.$e$.,$ areal fraction in $2$D$).$

$($c$)$ Estimate the average first$-$neighbor spacing $($use center$-$to$-$center distances$).$

$($d$)$ Estimate the average second$-$neighbor spacing.

$($e$)$ Estimate the average number density $($number of spheres per unit area$).$

$($f$)$ Select one sphere as the origin, and measure intersphere spacings out to at least $6R_o$ from

the origin. Draw a histogram illustrating the number of spheres vs$.$ spacing using $dr=$

$0\mathrm{.}2R_o$ increments for the spacing.

$($g$)$ Make a plot of the pair distribution function $g(r)$ for this structure using $dr=0\mathrm{.}2R_o.$

$($h$)$ Estimate $.$

$($i$)$ Discuss how your results from parts $($a$)$ and $($g$)$ compare, and how you can improve your

analysis.

Hint: to identify sphere $($or disk$)$ locations and spacing, you can use free software such as

ImageJ and use "Analyze particles" tool.